12.1 Linear Programming Flashcards
Linear programming
- When faced with two or more limiting factors then the situation is more complicated and linear programming techniques must be used
- Limiting factors are called constraints
- This technique requires the translation of a decision problem into a system of variables, equations and inequalities
Step by step technique for maximum contribution
Step 1 - Define the variables - Simplify the equations by using abbreviations for products or variables (x, y)
Step 2 - State the objective function - Expresses the total contribution (maximum) that will be made from the production of two products (SPx + SPy)
Step 3 - State the constraints as an equation - The total for resource must be less than or equal to total available for the resource (Hx + Hy <= Avail Hrs) plus the non-negativity constraint (x, y >=0)
Step 4 - Draw the graph - Constraint lines can then be plotted on graph by finding two points on each line and joining them together (intersects on x and y axis) - this gives us the feasible region of production values (area withing all of constraint lines)
Step 5 - Find the optimal solution within feasible region. Two approaches:
* Using simultaneous equations
* Using an iso-contribution line
Minimisation problems
- Involve finding solutions to minimise costs
- The aim would therefore be to find a total cost line touching the feasible area at a tangent as close to origin as possible (instead of furthest)
Binding constraints
Are the constraints that are fully used at the optimal solution point, ie the maximum availability of the resource related to constraints is being used (resource used = resource available)
Non binding constraints
Are the constraints which are not fully used at the optimal solution point, and the resource being used is less that the resource available (those which do not pass through the optimal point)
Slack
- This is the amount of a resource which is under-utilised when the optimum plan is implemented
- Ie any of the non binding constraints which have not used the maximum availability of the resource
- Important to know because unused resources can be put to another use, eg hired out
- If constraint is binding then slack will be zero (aka scarce resource)
Surplus
- This is the utilisation of a resource over and above a minimum
- Surpluses arise in minimisation of cost problems where there are greater than or equal to constraint lines
- If constraint is binding then surplus will be zero
The shadow price
The problem of the scarce resource could be alleviated by buying additional quantities at a premium price which would expand the feasible region (moving optimal solution point outwards)
- The maximum premium (over and above) on price that org would pay for extra resource is known as shadow price
- It is the extra contribution that would arise if one more unit of scarce resource became available or the drop in contribution as result of having one fewer unit of scarce resources
- Only applies to binding constraints
How to calculate shadow prices
Step 1 - Take equations of lines that intersect at optimal point, add one unit to constraint and leave other unchanged
Step 2 - Use simultaneous equations to calculate new optimal solution
Step 3 - Calculate the revised optimal contribution and compare to original, the increase is the shadow price
Minimum contractual requirements
If there is a customer order that the bus has to meet this needs to be taken into account before formulating the linear problem
Assumptions and Limitations of linear programming
- Linear relationships must exist
- It assumes there is a single quantifiable objective eg, maximise contribution (might b multiple)
- When there are a number of variables it becomes too complex and requires a computer to solve
- It is assumed that variables are completely divisible
- Single value estimates are used for the uncertain variables
- It is assumed that the situation remains static in all other respects
- It is assumed that the scenario is short term which allows us to ignore fixed costs
Linear programming - Spreadsheets - Advantages:
- Relatively easy to use
- Spreadsheet functions enable data to be processed more quickly
- More complex scenarios with more data can be worked with
- They include graphing functions that allow for quick reporting and analysis of data
- They should reduce calculation errors
- Formulae and look up tables can be used so that if any figure is amended, all the figures will be immediately recalculated which is useful for sensitivity analysis
- Results can be printed out or distributed to other users electronically, quickly and easily
Linear programming - Spreadsheets - Disadvantages:
- There can be sharing violations among users wishing to view or change data at same time
- They are not able to identify data input errors and error can impact validity of all data
- They are open to cyber attack through viruses, hackers and general system failure
- They do not take into account qualitative factors related to decisions which are equally important
- Spreadsheets do not make the overall decision
- Simple off the shelf programmes may not have full capabilities to solve all problem scenarios therefore more advanced (expensive) may be needed for org who expect to make decisions more often
Data visualisation
- Spreadsheets can be supplemented with charts, diagrams and pictures to help visualise problems and solutions which can make decisions on optimal mix less challenging
- Advantage is that it’s easier to recognise trends and patterns more quickly and easier to understand and digest this info
Non financial data in product mix decisions
- Customer value placed on product
- Competitor plans for competing products
- The length of time that a particular resource will be scarce
- Opportunities to use alternative resources through product redesign
- Expected demand for a product
- The ethical impacts of removing a product
